Theorem nnssn0s 28689

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28683	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4083	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3977	1 ⊢ ℕs ⊆ ℕ0s

This proof depends on syntax axioms:   ∖ cdif 3896   ⊆ wss 3899  {csn 4584   0_s c0s 28173  ℕ_0scn0s 28680  ℕ_scnns 28681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-ss 3916  df-nns 28683

This theorem is used by:  nnssno  28690  nnn0s  28695  nnn0sd  28696  nnsgt0  28707